On the basis of the three‐band Anderson lattice model for the CuO2 layers in high‐Tc superconductors we derive an effective Cu‐spin O‐hole Hamiltonian which yields simultaneously superexchange and RKKY interaction. By means of two nested canonical transformations of the Schrieffer‐Wolff type we perform a perturbative algorithm up to the fourth order in the Cu–O hybridization amplitude. Both O bands are taken into account consequently. The total indirect exchange integral reflects, in dependence on range and O‐hole doping, magnetic frustration and the competition between the super and RKKY exchanges of the Cu spins.
he low-energy Hamiltonian lor localized Cu spins and mobile 0 holes is rederived from the minimal version of the Emery model ascribed to a CuOi layer in high.?: superconductors. By means of a mmpulalional algolithm the SchrieRer-Woiff transformation is performed. at large but finite Hubbard wpulsion U on CU sites, up to fourth order in the d-p (Cu-0) hopping amplitude t . ?he resultant interactions in O(t') involve spinspin superexchange, a Kondo-like spin-hole exchange opposite in sign 10 that in O(f2). and a Huhbard.type attraction ktween symmetlized 0 holes within a CuO, gmup. An effective tpp-Jdd-Jdp-Jpp model is esfablished for the special case U = 2c, where L is the charge wdnsfer energy lntmductionThe electronic properties of a Cu-0 plane, as a basic structural element of the high-Tc superconductors La,_,Sr,CuO, and Y Ba,Cu,O,-,, can be described by the Emery model [I]. Such an approach allows us to distinguish the role of the copper (Cu) dstates and oxygen (0) p-states by assuming that strong electronic correlations exist (at least on Cu sites) and holes added by doping go predominantly to 0 sites. Treating the CU-0 hybridization perturbatively, a canonical transformation from the original Hamiltonian to an approximate effective one brings out the low-lying exitations in the CuO, layers. The Schrieffer-Wolft' transformation (21 and other perturbation techniques (as used, e.g., in [3, 41) have been applied to derive from the Emery model, or from the Anderson lattice model, a lot of spin exchange models [3-191. The transformed models include local Cu-spins and mobile 0-holes in the spin-fluctuation regime [3]; for the charge-fluctuation regime cf. [19]. The fourth-order contribution with respect to the d-p hopping amplitude, 1, contains, as a standard result [ZO], the Cu-Cu superexchange of Heisenberg type. Additionally, there must occur [3] spin-hole, spin-spin-hole, hole-hole-spin, and hole-hole interactions in O(t4). In particular, a carrier-carrier attraction was obtained in [3]; Cu-0 Kondo-like couplings with different signs arise from O ( t 4 ) in [4]. Some ambiguities in O(f4) concerning the structure of the interactions and the renormalization of the coupling parameters have motivated us to reconsider the problem. The aim is to give a straightforward microscopic foundation of an effective model Hamiltonian for describing the interplay between magnetism and superconductivity i n the Cu-spin and 0-hole subsystems. Usually, the single-band strongly correlated
An effective Hamiltonian for localized Cu spins and mobile 0 holes is deduced from the three-band Anderson lattice model ascribed to a single CuO, layer in high-T, superconductors. At large but finite Hubbard repulsion strength U on Cu sites the perturbative approach is given completely up to the fourth order in the Cu(d)-O(p) hybridization amplitude t. Both the Schrieffer-Wolff transformation and the projection onto the subspace of singly occupied Cu sites are performed by means of a computational algorithm. The resultant interactions in O(t") include a Kondo-like spin-hole exchange in competition to that in O(t')), the spin-spin superexchange of the Heisenberg-type, spin-spin-hole, hole-hole, and spin-hole-hole correlations. The spin-hole-hole coupling, reflecting p-hole hopping affected by a Kondo exchange, is due to the direct 0-0 transfer t, inherent in the periodic Anderson model. It is shown how in the limit t, = 0 the effective Hamiltonian reduces to that originated from the minimal version of the Emery model. Aus dem Dreiband-Anderson-Gitter-Modell fur eine Cu0,-Schicht in Hoch-T,-Supraleitern wird ein effektiver Hamilton-Operator fur lokalisierte Cu-Spins und mobile 0-Ldcher hergeleitet. Bei groRer, aber endlichcr Starke Li der Hubbard-AbstoRung an Cu-Platzen wird der storungstheoretische Zugang bis zur vierten Ordnung in der Cu(d)-O(p)-Hybridisierungsamplitude t vollstandig angegeben. Die Schrieffer-Wolff-Transformation und die Projektion auf den Unterraum der einfach besetzten Cu-Platze werden mit Hilfe eines Computer-Algorithmus durchgefiihrt. Die in O(t4) resultierenden Wechselwirkungen enthalten einen Kondo-ahnlichen Spin-Loch-Austausch konkurrierend zu dem in O(tZ), den Spin-Spin-Superaustausch vom Heisenberg-Typ, Spin-Spin-Loch-, Loch-Loch-und Spin-Loch-Loch-Korrelationen. Die Spin-Loch-Loch-Kopplung, die ein durch Kondo-Austausch beeinfluotes p-Loch-Hupfen widerspiegelt, folgt aus dem direkten 0-0-Transfer t , im periodischen Anderson-Modell. Es wird gezeigt, wie sich im Grenzfall t, = 0 der effektive Hamilton-Operator auf den aus der minimalen Version des Emery-Modells stammenden reduziert. ') Linnestr. 5, D-04 109 Leipzig, Federal Republic of Germany. ' ) Hans-Otto-Str. 5/707,
On the basis of a n extended attractive Hubbard model with nearest-neighbour interaction of opposite spins two functional methods are adopted for deriving a self-consistent equation for the superconducting order parameter. The functional-integral formalism is exploited in saddle-point approximation. The functional-derivative technique using Schwinger source fields leads via the self-energy in Hartree-Fock-Bogolubov approximation t o a nonlinear gap equation. The superconducting transition temperature is investigated for the case of s-wave pairing.Auf der Grundlage eines erweiterten anziehenden Hubbard-Modells mit einer Wechselwirkung benachbarter Elektronen entgegengesetzten Spins werden zwei funktionale Methoden getestet, um Selbstkonsistenzgleichungen fur den supraleitenden Ordnungsparameter abzuleiten. Der Funktional-Integral-Formalismus wird in Sattelpunktsnaherung behandelt. Die BunktionalAbleitungs-Technik, die Schwingersche Quellfelder verwendet, fuhrt uber die Selbstenergie in Hartree-Fock-Bogolubov-Approximation zu einer nichtlinearen Gap-Gleichung. Die supraleitende ubergangstemperatur wird fur den Fall der s-Wellenpaarung untersucht.
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